Chapter 1 will explain what this book is about. Here I will explain why I chose to write the book, how it is written, when and where the work was done, and who helped.

Why. It would make a good story if I was inspired to write this book by an image of Paul Erdös magically appearing on a cheese quesadilla, which I later sold for thousands of dollars on eBay. However, that is not true. The three main events that led to this book were (i) the use of random graphs in the solution of a problem that was part of Nathanael Berestycki's thesis; (ii) a talk that I heard Steve Strogatz give on the CHKNS model, which inspired me to prove some rigorous results about their model; and (iii) a book review I wrote on the books by Watts and Barabási for the Notices of the American Math Society.

The subject of this book was attractive for me, since many of the papers were outside the mathematics literature, so the rigorous proofs of the results were, in some cases, interesting mathematical problems. In addition, since I had worked for a number of years on the properties of stochastic spatial models on regular lattices, there was the natural question of how the behavior of these systems changed when one introduced long-range connections between individuals or considered power law degree distributions.

In this chapter we will introduce and study the random graph model introduced by Erdös and Rényi in the late 1950s. This example has been extensively studied and a very nice account of many of the results can be found in the classic book of Bollobás (2001), so here we will give a brief account of the main results on the emergence of a giant component, in order to prepare for the analysis of more complicated examples. In contrast to other treatments, we mainly rely on methods from probability and stochastic processes rather than combinatorics.

To define the model, we begin with the set of vertices V = {1, 2, … n}. For 1 ≤ x < y ≤ n let ηx,y be independent = 1 with probability p and 0 otherwise. Let ηy,x = ηx,y. If ηx,y = 1 there is an edge from x to y. Here, we will be primarily concerned with situation p = λ/n and in particular with showing that when λ < 1 all of the components are small, with the largest O(log n), while for λ > 1 there is a giant component with ~ g(λ)n vertices. The intuition behind this result is that a site has a Binomial(n – 1, λ/n) number of neighbors, which has mean ≈ λ.

Definitions and Heuristics

In an Erdös–Rényi random graph, vertices have degrees that have asymptotically a Poisson distribution. However, as discussed in Section 1.4, in social and communication networks, the distribution of degrees is much different from the Poisson and in many cases has a power law form, that is, the fraction of vertices of degree k, pk ~ Ck-β as k → ∞. Molloy and Reed (1995) were the first to construct graphs with specified degree distributions. We will use the approach of Newman, Strogatz, and Watts (2001, 2002) to define the model.

Let d1,…dn be independent and have P(di = k) = pk. Since we want di to be the degree of vertex i, we condition on En = {d1 + … + dn is even}. If the probability P(E1) ∊ (0, 1) then P(En) → ½ as n → ∞ so the conditioning will have little effect on the finite-dimensional distributions. If d1 is always even then P(En) = 1 for all n, while if d1 is always odd, P(E2n) = 1 and P(E2n+1) = 0 for all n.

To build the graph we think of di half-edges attached to i and then pair the half-edges at random. The picture gives an example with eight vertices.

Introduction to the Introduction

The theory of random graphs began in the late 1950s in several papers by Erdös and Rényi. However, the introduction at the end of the twentieth century of the small world model of Watts and Strogatz (1998) and the preferential attachment model of Barabási and Albert (1999) have led to an explosion of research. Querying the Science Citation Index in early July 2005 produced 1154 citations for Watts and Strogatz (1998) and 964 for Barabási and Albert (1999). Survey articles of Albert and Barabási (2002), Dorogovstev and Mendes (2002), and Newman (2003) each have hundreds of references. A book edited by Newman, Barabási, and Watts (2006) contains some of the most important papers. Books by Watts (2003) and Barabási (2002) give popular accounts of the new science of networks, which explains “how everything is connected to everything else and what it means for science, business, and everyday life.”

While this literature is extensive, many of the papers are outside the mathematical literature, which makes writing this book a challenge and an opportunity. A number of articles have appeared in Nature and Science. These journals with their impressive impact factors are, at least in the case of random graphs, the home of 10 second sound bite science.

Barabási–Albert Model

As we have noted, many real-world graphs have power law degree distributions. Barabási and Albert (1999) introduced a simple model that produces such graphs. They start with a graph with

Bollobás, Riordan, Spencer, and Tusnády (2001) complain: “The description of the random graph process quoted above is rather imprecise. First as the degrees are initially zero, it is not clear how the process is supposed to get started. More seriously, the expected number of edges linking a new vertex to earlier vertices is ΣiΠ(ki) = 1, rather than m. Also when choosing in one go a set S of m earlier vertices as neighbors of v, the distribution is not specified by giving the marginal probability that each vertex lies in S.”

As we will see below there are several ways to make the process precise and all of them lead to the same asymptotic behavior.

The Lion and the Christian. A lion and a Christian in a closed circular Roman arena have equal maximum speeds. What tactics should the lion employ to be sure of his meal? In other words, can the lion catch the Christian in finite time?

Integer Sequences

1. (i) Show that among n + 1 positive integers none of which is greater than 2n there are two such that one divides the other.

2. (ii) Show that among n + 1 positive integers none of which is greater than 2n there are two that are relatively prime.

3. (iii) Suppose that we have n natural numbers none of which is greater than 2n such that the least common multiple of any two is greater than 2n. Show that all n numbers are greater than 2n/3.

4. (iv) Show that every sequence of n = rs + 1 distinct integers with r, s ≥ 1 has an increasing subsequence of length r + 1 or a decreasing subsequence of length s + 1.

Points on a Circle

1. (i) Let X and Y be subsets of the vertex set of a regular n-gon. Show that there is a rotation ϱ of this polygon such that |X ∩ ϱ(Y)| ≥ |X||Y|/n, where, as usual, |Z| denotes the number of elements in a finite set Z.

2. […]

When I was putting together this collection of problems, I always asked myself whether the two giants of mathematics I had the good fortune to know well, Paul Erdős and J.E. Littlewood, would have found the question interesting. Would they have felt enticed to think about it? Could they have not thought about it, whether they wanted to or not? I think that many of the problems that ended up in this volume are indeed of the kind Erdős and Littlewood would have found difficult not to think about; since this collection contains many problems they considered or even posed, this assertion may not be as preposterous as it seems.

I was not yet ten when I fell in love with mathematical problems. Growing up in Hungary, this love got plenty of encouragement, and when at fourteen I got to know Paul Erdős, the greatest problem poser the world has ever seen, my fate was sealed. He treated me and other young people to a variety of beautiful and fascinating problems, solved and unsolved; many of the solved ones I heard from him in my teens appear in this volume.

The impetus for putting together this collection of problems came much later, in Memphis, where, for a few years now, some of the local and visiting mathematicians have had the habit of having lunch together, followed by coffee and a mathematical problem or two in my office.

The Lion and The Christian

A lion and a Christian in a closed circular Roman arena have equal maximum speeds. Can the lion catch the Christian in finite time?

Solution. At the first sight the Hint gives an elegant and very simple solution. Indeed, writing O for the centre of the arena, L for the lion, and M (‘man’) for the Christian, if L keeps on OM and approaches M at maximal speed then we may simplify the calculations by making M run along the boundary circle of radius 1. Then if L starts at the centre (which may clearly be assumed) then L will run along a circle of radius 1/2, so L will catch M in the time it takes to cover distance π. This assertion is instantly justified by Figure 22 which shows that if the arc length MM′ on the outer circle of radius 1 is φ then OPL′ is also φ and hence OSL′ is 2φ, where S is the centre of the inner circle (of radius 1/2) touching the outer circle in P and the line OM in O. Consequently, in the time it takes the man to get from M to M′ on the boundary circle, the lion gets from L = O to L′. Hence L catches M in P. (Equivalently, the angles marked ψ and 2ψ show that the arc length M′ P on the boundary circle is precisely the arc length L′P on the inner circle.)

1. Let O be the centre of the circle, L the lion and M the Christian. What happens if L keeps on the radius OM and approaches M at top speed?

3. What about a random rotation?

For the second part, use a suitable approximation.

4. Can the line be partitioned into countably many closed sets?

5. Try local changes.

7. Imagine that you can use negative amounts of fuel as well (i.e., can get a loan for your future intake), and every time you get to a town, you get a new lot of fuel. What happens if you go round and round the circuit?

10. Ask the same question for a ‘suitable’ countable set.

11. Which sums seem to be most likely to come up, and which ones seem least likely?

14. Show that the following greedy algorithm constructs a sufficiently large independent set. Pick a vertex x1 of minimal degree in G1 = G, and let G2 be the graph obtained from G1 by deleting x1 and its neighbours. Pick a vertex x2 of minimal degree in G2, and let G3 be obtained from G2 by removing x2 and its neighbours. Proceed in this way, stopping with Gℓ and xℓ, when Gℓ is a complete graph.

A convenient fact which was observed and then repeatedly used in the preceding lectures is that one can construct a family of noncommutative random variables with a prescribed joint distribution (where quite often the joint distribution is indicated in the guise of a prescribed joint R-transform). Sometimes there may be more than one way of doing such a construction; an example of such a situation is the one of free semicircular families, which can be obtained by an abstract free product construction, but can also be “concretely” put into evidence by using operators of creation and annihilation on the full Fock space (cf. Lecture 7).

Of course, all the different methods for constructing a family of elements with a prescribed joint distribution are ultimately equivalent, in the respect that the calculations with moments and with cumulants performed on the family give the same results, no matter how the family was constructed. Nevertheless, there can be a substantial difference in the transparency of the calculations – it may happen that the solution to the problem we are trying to solve shows up more easily if one method of construction is used over another.

So this is, in a nut-shell, the idea of “modeling”: find a good way of constructing a family of non-commutative random variables with a given joint distribution, so that we are at an advantage when computing moments and cumulants of that family.